tetration from alternative fixed point

[tommy1729]

If I compute the Abel function, I dont get singularities at 1,e,e^e,... for the secondary fixed point.
I use something similar to the formula:
\( \text{slog}(z)=\log_c(\log_\ast^{[n]}(z)-e[2])-n \) where c is the derivative at the secondary fixed point e[2] and \( \log_\ast \) is the branch of the logarithm with imaginary part between \( 2\pi \) and \( 4\pi \), which implies that \( \log_\ast^{[n]}(z)\to e[2] \).

But with this formula repeated application of \( \log_\ast \) to 1 is possible without running into 0.
So no singularities at 1,e,e^e,etc.
How do you do it?

is that slog(z) analytic ?

is the related superfunction a superfunction of exp(z) + 2pi i then ?

but then it cant be related to the superfunction of exp(z) , can it ?